2020 AMC 12B Problems/Problem 6
Problem
For all integers the value of is always which of the following?
Solution 1
We first expand the expression:
We can now divide out a common factor of from each term of this expression:
Factoring out we get
which proves that the answer is
Solution 2
In the numerator, we factor out an to get Now, without loss of generality, test values of until only one answer choice is left valid:
- knocking out and
- knocking out
This leaves as the only answer choice left.
This solution does not consider the condition The reason is that, with further testing it becomes clear that for all we get as proved in Solution 1. The condition was most likely added to encourage the selection of choice
~DBlack2021 (Solution Writing)
~Countmath1 (Minor edits in formatting)
~MRENTHUSIASM (Edits in Logic)
Video Solution
https://youtu.be/ba6w1OhXqOQ?t=2234
~ pi_is_3.14
Video Solution
~IceMatrix
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.